Rubik cube sequence?

[Brian L. Galebach]

I don't think I said that right.  What I'm trying to say is that the 10th
term of the sequence must exclude positions that are reachable in 9 moves or
less.  Therefore, some positions that are reachable within 10 moves should
not be included in the total for the 10th term.  (11th term since 0 moves is
the first term.)  So how would one say this?  Perhaps "Positions reachable
in n moves, but not reachable in less than n moves."  Kind of wordy.

Brian

-----Original Message-----
From: Brian L. Galebach [mailto:briang at segmail.com]
Sent: Friday, February 21, 2003 5:33 PM
To: seqfan at ext.jussieu.fr
Cc: briang at ProbabilitySports.com
Subject: RE: Rubik cube sequence?


Hello,

I can't tell by quick inspection of the terms given so far, but I just
wanted to make sure you were calculating the same thing as in A079761.  That
sequence gives the number of positions that are n moves away from being
solved.  Translated another way would be "positions reachable in at most n
moves".  Therefore, the 100th term of the sequence, for example, would be 0,
whereas "reachable in n moves" or "reachable within n moves" would give the
total number of combinations in the cube.

Brian

-----Original Message-----
From: Alex Healy [mailto:ahealy at fas.harvard.edu]
Sent: Friday, February 21, 2003 5:13 PM
To: seqfan at ext.jussieu.fr
Cc: njas at research.att.com
Subject: RE: Rubik cube sequence?


As promised, here is the (beginning of the) sequence for the 3x3x3 cube.
Again, half-twists count as a move, so there are 18 positions reachable in 1
move, etc.  The distinction between positions reachable "in n moves" and
positions reachable "within n moves" vanishes after 2 moves because there is
a cycle of length 3, e.g. (R, R^2, R).  Here's the sequence:

1, 18, 262, 3502, 46741, 621649, 8240087

I was having trouble with memory usage, but maybe with a little tweaking
I'll be able to get another term or so.  This is it for now, though.

Alex


> -----Original Message-----
> From: N. J. A. Sloane [mailto:njas at research.att.com]
> Sent: Friday, February 21, 2003 1:50 PM
> To: seqfan at ext.jussieu.fr
> Subject: Rubik cube sequence?
>
>
> Inspired by Jaap Scherphuis's excellent web site
> on generalizations of Rubik's cube,
> i've added several sequences to the OEIS giving the number
> of positions that are n moves away from the start in
> these puzzles.  For example:
>
> %I A079761
> %S A079761
> 1,9,54,321,1847,9992,50136,227536,870072,1887748,623800,2644
> %N A079761 Number of positions that are n moves from the
> starting position in the 2 X 2 X 2 Rubik cube puzzle.
> %C A079761 A puzzle in the Rubik cube family. The total
> number of distinct positions is 3674160. A half-turn is
> considered to be one move.
> %D A079761 D. R. Hofstadter, Metamagical Themas, Basic Books,
> NY, 1985, p. 359.
> %H A079761 Jaap Scherphuis, <a
> href="http://www.geocities.com/jaapsch/puzzles/">Puzzle Pages</a>
> %K A079761 nonn,fini,full,new
> %O A079761 0,2
> %A A079761 njas, Feb 20 2003
>
> Does anyone know the analogous sequence for the 3 X 3 X 3 cube,
> or even the beginning of that sequence?
>
> NJAS
>